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Previously, we have shown that the transition probability of the Landau-Zenerproblem in periodic lattice systems becomes large by taking into account thenonlinearity of the energy spectra, compared with the probability by theconventional Landau-Zener formula. The enhancement comes from the nonlinearitypeculiar to the periodic lattice system, and this effect from the lattice ontransition action cannot be neglected in the transition process.  In the present paper, we first give a brief review of the previous work, andconstruct the transfer matrix of the Landau-Zener problem by the semiclassicaldescription for lattice systems. Next, we study a ladder lattice system andshow that the transition action obtains a phase due to the nonlinearity. Then,we consider the double-passage problem of the ladder system within thesemiclassical description. We find the oscillation of the probability by thetransition phase by the lattice effect. This phase comes from the semiclassicalanalysis unlike the Stokes phase, and we show that the oscillation is mainlycontributed by the transition phase by the lattice effect, when thehybridization of the ladder is strong.

Semiclassical Lattice effects on interband tunneling of a two-state system